Questions: Enola has x quarters and y dimes. She has a minimum of 18 coins worth at most 3.60 combined. Solve this system of inequalities graphically and determine one possible solution.

Solution Steps

Enola has \( x \) quarters and \( y \) dimes. We need to set up the inequalities based on the given conditions:

1. She has a minimum of 18 coins.
2. The total value of the coins is at most $3.60.

The value of a quarter is $0.25 and the value of a dime is $0.10.

1. The total number of coins is at least 18: \[ x + y \geq 18 \]
2. The total value of the coins is at most $3.60: \[ 0.25x + 0.10y \leq 3.60 \]

To solve the system graphically, we need to plot the inequalities on a coordinate plane.

1. For \( x + y \geq 18 \):

   • Rearrange to \( y \geq 18 - x \).

2. For \( 0.25x + 0.10y \leq 3.60 \):

   • Rearrange to \( y \leq \frac{3.60 - 0.25x}{0.10} \).
   • Simplify to \( y \leq 36 - 2.5x \).

Find a point that satisfies both inequalities. For example, let's check \( x = 10 \) and \( y = 10 \):

1. Check \( x + y \geq 18 \): \[ 10 + 10 = 20 \geq 18 \quad \text{(True)} \]

2. Check \( 0.25x + 0.10y \leq 3.60 \): \[ 0.25(10) + 0.10(10) = 2.50 + 1.00 = 3.50 \leq 3.60 \quad \text{(True)} \]

Final Answer